statistical quality control telephone

Quality Control 2 Midterm

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STATISTICAL QUALITY CONTROL

Quality Control 2

College of Pharmacy

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STATISTICAL QUALITY CONTROL

Dr. Walter Shewhart (1920’s, Bell

Telephone Laboratories)

Introduced the concept of “controlling the

quality” rather than inspecting it into the

product.

Devised the Shewhart control chart technique

for in-process manufacturing operations

Introduced the concept of statistical sampling

inspection.

2

HISTORY

Pioneered by Walter

W. Edward Deming applied it in US during

WWII = Succeefully improved quality

Deming introduce to Japanese industry

(after the war has ended) = high quality

Japanese products

Shewart crated the basis and concept of

SPC 3

The quality of a manufactured product is

defined as its conformity to given

standards of specifications.

When measured, quality is always subject

to a certain amount of variation.

Variation is present in any process, deciding

when the variation is natural or when it

needs correction is the key to quality control

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SOURCES OF VARIATION

The two general causes of variation are:

1. Chance, or common

Inevitable or unavoidable, usually large in number and random in nature

e.g. slight differences in process variables like diameter, weight, service time, temperature

2.Assignable

- nonrandom, can be identified and eliminated e.g. poor employee training, worn tool, machine needing

repair

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To maintain quality at optimum level with a

minimum of non-uniformity, quality control

is faced with the problem of

Evaluating the magnitude of chance variation

Detecting the existence of assignable causes

of variation which can be corrected.

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2

Statistical methodology, which is now an

integral part of many quality control

systems, has been shown to be applicable

in solving this problem.

These charts indicate to the workers or

group leaders whether the production is in-

control status or unsatisfactory production

when out-of-control.

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Statistical quality control

is the monitoring of quality by the application of statistical methods in all stages of production. – Descriptive statistics

it consists of proper sampling, determining quality variation of the sample, and making inferences to the entire batch under investigation. - Acceptance sampling inspection

It makes use of control charts, a tool which may influence decisions related to the functions of specification, production or inspection. – Statistical process control

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TRADITIONAL STATISTICAL TOOL

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• Descriptive Statistics include

– The Mean- measure of central tendency

– The Range- difference between largest/smallest observations in a set of data

– Standard Deviation measures the amount of data dispersion around mean

n

x

x

n

1i

i

1n

Xx

σ

n

1i

2

i

OTHER STATISTICAL TOOLS

Mode

• the most commonly occurring value

ex: 6 people with ages 21, 22, 21, 23, 19, 21 - mode = 21

Median

• the center value

• the formula is N+1/2

ex: 6 people with ages 21, 22, 24, 23, 19, 21

line them up in order form lowest to highest

19, 21, 21, 22, 23, 24

and take the center value - mode =21.5

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VARIABLES

Categorical (aka Qualitative)

takes on values that are names or labels

Quantitative (aka Numerical)

represent a measurable quantity

Examples:

Color of the ball

Population number in Metro Manila

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VARIABLES

Independent Variable

is the variable that is changed or controlled in a scientific experiment to test the effects on the DV ( treatment and controls)

Dependent Variable

is the variable being tested and measured scientifc experiment (result of the experiment)

Examples:

A scientist wants to see if the brightness of light has any effect on moth being attracted to the light.

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3

DRY MIX

Dependent Variable

Responding Variable

Y-axis (vertical axis)

Manipulated variable

Independent variable

X-axis (Horizontal axis)

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QUALITY CONTROL CHARTS

There are two basic quality control

charts, which are based on the

measurability of the quality characteristics,

namely:

1. Variable Chart – This is a chart using

actual records of numerical measurement

on a full continuous scale such as meter,

grams, and liter. Examples of variable

charts are the X (mean) and R (range)

charts.

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The average (mean) is the most used

in quality control chart, while the range

and standard deviation are used as

measures of dispersion.

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2. Attribute Chart – This is a chart, which makes use of discrete data classifying the number of items conforming and the number of items failing to conform to any specified requirements. An example of an attribute chart is the control chart for fraction defective known as P chart. Another type is c chart which shows the number of defects per unit.

It is also known as a go or no-go chart. Graphically, these charts show the proportion of the production or per unit that is not acceptable.

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A control chart consists of a solid line and

two horizontally parallel lines on either

side of the solid line. The control solid

line is the target value of the historical

process average and/ or range.

The two dotted parallel lines indicate

the limits within which practically all

observations should fall as long as the

process is under normal variation or

known as “statistically controlled”.

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4

The upper dotted line is the upper control limit (UCL) and the lower line is the lower control limit (LCL), both of these are three standard deviations above and below, respectively, the central line.

The six standard deviations, if the process is in control, spread between the upper and lower control limits encompass 99.7 % of the values in a normal distribution with its mean at the central line.

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In statistics, the 68–95–99.7 rule, also known as the three-sigma rule or empirical rule, states that nearly all values lie within three standard deviations of the mean in a normal distribution

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The control limits on the chart are so

placed as to disclose the presence or

absence of assignable causes. Although

their actual elimination is usually an

engineering job, the control chart tells

when, and in some instances, suggests

where to look.

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• Control Charts show sample data plotted on a graph with CL, UCL, and LCL

• Control chart for variables are used to monitor characteristics that can be measured, e.g. length, weight, diameter, time

• Control charts for attributes are used to monitor characteristics that have discrete values and can be counted, e.g. % defective, number of flaws in a shirt, number of broken eggs in a box

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STATISTICAL CONTROL OF QUALITY CHARACTERISTICS

The principle of the control chart technique is that quality measurements obtained from samples from production will vary due to chance causes or assignable causes. When all observations are found within the limits, the process is in control. If an observation is found outside the limit lines, this variation is due to an assignable cause.

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General Method:

1. Select a sample size, (n) at random from

the production.

2. Compute an average for each set of

sample measurements. MEAN

3. Compute the appropriate standard

deviation of the average used. SD

4. From the computed standard deviation,

compute for the UCL and LCL.

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5. Prepare the control chart by drawing a solid horizontal line extending from the vertical quality scale at the average value. A pair of dotted lines or broken lines (control limits) are drawn on either side of this central line at a distance x times the standard deviation.

6. Plot the averages obtained from the sample average values. If any of the plotted points fall outside of the established control limits, the process is out-of control.

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Figure 3-1. A control chart for variables

VARIABLE CHARTS

Variables refer to characteristics that can be

measured. Measurements of parts may vary in

length, diameter, tensile strength and so on.

Mean chart – the first chart designed for

variable with the purpose that is to portray the

fluctuation in the sample means. The mean of

these sample means is denoted as double bar x

or the mean of means.

Range Chart – shows variation in the ranges of

the samples.

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Use x-bar charts to monitor the changes in the mean of a process (central tendencies)

Use R-bar charts to monitor the dispersion or variability of the process

System can show acceptable central tendencies but unacceptable variability

Use the two charts together

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6

CONSTRUCTING A X-BAR CHART: A QUALITY CONTROL INSPECTOR AT LIQUID SECTION HAS

TAKEN FOUR SAMPLES WITH THREE OBSERVATIONS (HOURLY) EACH OF THE VOLUME OF

BOTTLES FILLED. IF THE STANDARD DEVIATION OF THE BOTTLING OPERATION IS .2 OUNCES, USE

THE BELOW DATA TO DEVELOP CONTROL CHARTS WITH LIMITS OF 3 STANDARD DEVIATIONS FOR

THE 16 OZ. BOTTLING OPERATION.

xx

xx

n21

zσxLCL

zσxUCL

sample each w/in nsobservatio of# the is

(n) and means sample of # the is )( where

n

σσ ,

...xxxx x

kk

Hour 1 Hour 2 Hour 3

Sample 1 15.8 16.1 16.0

Sample 2 16.0 16.0 15.9

Sample 3 15.8 15.8 15.9

Sample 4 15.9 15.9 15.8

Sample means (X-bar)

Sample ranges (R)

© Wiley 2007

Center line and control limit

formulas

SOLUTION AND CONTROL CHART (X-

BAR)

Center line (x-double bar):

Control limits for±3σ limits:

15.923

15.915.97515.875x

15.624

.2315.92zσxLCL

16.224

.2315.92zσxUCL

xx

xx

© Wiley 2007

Sum of the means of the subgroup (Samples)

X = ----------------------------------------------------- Number of the sample means

Compute for the standard error of the distribution of sample means designated and found by

δ

δx = ----------------------

Square root of n

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= =

UCL = X + 3 δ LCL = X - 3δ

---- --

square root of n square root of n

Or

= _ = _

UCL = X + A2 R LCL = X - A2R

=

Where: X = mean of the sample means

_

R = mean of the ranges

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Observations

in Sample

Size

(n)

Factors for

Mean Chart

(X)

Factors for Range

Chart

(R)

A2 D3 D4

2 1.88 0.00 3.27

3 1.02 0.00 2.57

4 0.73 0.00 2.28

5 0.58 0.00 2.11

6 0.48 0.00 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10 0.31 0.22 1.78 Use Table 3-2 to estimate the 3 standard deviation

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SECOND METHOD FOR THE X-BAR CHART

USING

R-BAR AND THE A2 FACTOR (TABLE 6-1)

Use this method when sigma for the process

distribution is not know

Control limits solution:

______ .2330.7315.91RAxLCL

______.2330.7315.91RAxUCL

.2333

0.20.30.2R

2x

2x

© Wiley 2007


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CONTROL CHART FOR RANGE (R)

0.00.0(.233)RDLCL

.532.28(.233)RDUCL

.2333

0.20.30.2R

3

4

R

R

© Wiley 2007

Center Line and Control Limit

formulas:

Observations

in Sample

Size

(n)

Factors for

Mean Chart

(X)

Factors for Range

Chart

(R)

A2 D3 D4

2 1.88 0.00 3.27

3 1.02 0.00 2.57

4 0.73 0.00 2.28

5 0.58 0.00 2.11

6 0.48 0.00 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

9 0.34 0.18 1.82

10 0.31 0.22 1.78 Use Table 3-2 to estimate the 3 standard deviation

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Problem Solving:

The volume of 5 vials was determined

during the filling of an injectable.

Determine the UCL and LCL using the

formulas below:

= _ = _

UCL = X + A2 R LCL = X - A2R

Or

_ _

UCL = D4 R LCL = D3 R

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No. Of Volume (ml) of 5 vials every 30 mins

Inspection A B C D E X R

1 10.5 10.3 10.2 10.6 10.7

2 10.4 10 10.3 10.2 10.1

3 10.1 9.8 9.9 9.6 9.7

4 9.8 9.8 10 10.2 10.1

5 10.6 10.5 10.5 10.5 10.4

6 10.7 10.5 10.6 10.7 10.9

7 10.7 10.3 10.5 10.4 10.4

8 10.8 10.2 10.7 10.8 10.9

9 10 10.3 10.4 10.1 10

10 10 10 10 10.4 10.3

11 9.8 9.9 9.8 9.8 9.9

12 10.7 10.6 9.6 10.9 10.5

13 10.1 10.6 10.4 10.7 10.3

14 10.5 10.9 10.5 10.6 10.4

15 10.3 10.3 10.2 10.5 10.3

X= R= 44

ATTRIBUTE CHARTS

Percent Defective Chart – also known

as P-chart or “p bar chart” – “p”. This chart

shows graphically the proportion of the

production that is not acceptable.

C Bar Chart - c chart portrays the

number of defects per unit, this is to show

graphically how many defects appear in a

unit of a production.

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To construct a control chart for fraction

defective the following steps are

performed:

1. Record the number inspected (n) and the

number of defectives found (d).

2. Compute for fraction defective (p) which

is the ratio of the number of defectives

found to the total number of units actually

inspected in the batch.

p = d

n 50


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3. Compute for the average fraction

defective (p) obtained by dividing the total

number of defectives found by the total

number of units inspected in a series of

batch.

p = ∑ d

∑ n

4. Calculate the UCL and LCL through the

following formulas:

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UCL and LCL = p ± 3

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n

)p(1p

5. Plot the p’s on the control chart with p on

the center line.

6. When all the points fall within the control

limits, the product is said to be in

statistical control.

Percent defective (100P), a more

convenient value may be used in

constructing the P chart.

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P-CHART EXAMPLE: A PRODUCTION MANAGER FOR A TIRE COMPANY HAS

INSPECTED THE NUMBER OF DEFECTIVE TIRES IN FIVE RANDOM SAMPLES

WITH 20 TIRES IN EACH SAMPLE. THE TABLE BELOW SHOWS THE NUMBER

OF DEFECTIVE TIRES IN EACH SAMPLE OF 20 TIRES. CALCULATE THE

CONTROL LIMITS.

Sample Number of

Defective Tires

Number of Tires in each

Sample

Proportion Defective

(fraction defective)

1 3 20 .15

2 2 20

3 1 20

4 2 20

5 2 20

Total

3(.067).1σzpLCL

3(.067).1σzpUCL

20

(.1)(.9)

n

)p(1pσ

100

10

Inspected Total

Defectives#pCL

p

p

p

Solution:

© Wiley 2007

Problem Solving:

A batch of ointment was filled into tubes

during ten working days. 500 tubes were

filled each day. The inspector withdraw a

random sample of based on the master

table and noted the number of leaking

tubes below.

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Day No. Of Leaking Tubes

1 4

2 6

3 7

4 5

5 3

6 1

7 6

8 3

9 2

10 0 56


9

Day n d p

1 4

2 6

3 7

4 5

5 3

6 1

7 6

8 3

9 2

10 0

∑n= ∑d= p=

57

a. Construct a control chart for the crimping

process.

b. Is the process statistically controlled?

c. If the AQL is 2.5, give the acceptance criteria

C-CHART EXAMPLE: THE NUMBER OF WEEKLY CUSTOMER

COMPLAINTS ARE MONITORED IN A LARGE HOTEL USING A C-

CHART. DEVELOP THREE SIGMA CONTROL LIMITS USING THE

DATA TABLE BELOW. NUMBER OF UNITS USED ARE PERIOD OF

TIME, SURFACE AREA OR VOLUME.

Week Number of Complaints

1 3

2 2

3 3

4 1

5 3

6 3

7 2

8 1

9 3

10 1

Total 22

02.252.232.2ccLCL

6.652.232.2ccUCL

2.210

22

samples of #

complaints#CL

c

c

z

z

Solution:

© Wiley 2007

The arithmetic mean number of defects

per tuner (c) is found by:

sum of the defects

c = --------------------------

total number of units

UCL and LCL = c ± 3 √ c

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Problem Solving:

Ten of the newly-designed containers

were inspected for defects; the following

numbers were found 8, 5, 6, 4, 3, 8, 8, 10,

9, 9. construct the c bar chart and plot the

unit defects.

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SEATWORK

Time 1 2 3 4 5

8 am 6.04 6.01 6.05 6.02 6.06

9 am 6.01 6.02 6.03 6.02 6.02

10 am 6.01 6.05 6.07 6.03 6.04

11 am 6.02 6.04 6.04 6.03 6.02

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QC Inspector checks five pieces of the output of a shearing

machine every hour. She measure and records each piece.

Design x bar chart, plot the important data for four hours.

Design a range chart. Interpret the charts.

PROCESS CAPABILITY

The ability of production process to meet or

exceed preset specifications

Product specification

- Preset ranges of acceptable quality characteristics

- Tolerance

- an allowance above or below the nominal value.

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MEASURING PROCESS CAPABILITY RATIO

specification width (USL –LSL)

Cp = ----------------------------------------

Process width (6σ)

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PROCESS CAPABILITY VALUES

Cp = 1 The process variability just meets specification, process is minimally capable

Cp ≤ 1 The process variability is outside the specification, process is not capable

Cp ≥1 The process variability fall within specification limits, process exceeds minimal capability

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65

20 25 30

Upper

specification

Lower

specification

Nominal

value

PROCESS CAPABILITY

Process is capable

Process distribution

Process is not capable

20 25 30

Upper

specification

Lower

specification

Nominal

value

Process distribution

PROCESS CAPABILITY CONFIRMING PROCESS CAPABILITY

(USL - µ , µ - LSL)

Cpk = ------------------------------

( 3σ , 3σ )

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Process Capability Index, Cpk, is an index that measures the potential for a process to generate defective outputs relative to either upper or lower specifications


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EXAMPLE 1: INTENSIVE CARE LAB

The intensive care unit lab process has an average

turnaround time of 26.2 minutes and a standard

deviation of 1.35 minutes.

The nominal value for this service is 25 minutes

with an upper specifications limit of 30 minutes

and a lower specifications of 20 minutes.

The administrator of the lab wants to have three-

sigma performance for her lab. Is the lab process

capable of this level of performance?

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ASSESSING PROCESS CAPABILITY

Upper Specifications: 30 minutes

Lower Specifications: 20 minutes

Average service: 26.2 minutes

σ: 1.35 minutes

specification width (USL –LSL)

Cp = ----------------------------------------

Process width (6σ)

Cp = 1.23 70


ASSESSING PROCESS CAPABILITY

Upper Specifications: 30 minutes

Lower Specifications: 20 minutes

Average service: 26.2 minutes

σ: 1.35 minutes

CpK = Minimum of

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3,

3

xficationupperspecificationlowerspecix

EXAMPLE 2

A bottling filling machine for Paracetamol syrup are evaluated for their capability

Specification is 15.8 – 16.2 ml and mean is 15.9

Which of the machines is capable?

72

Bottling machine Standard deviation

A .05

B 0.1

C 0.2

SAMPLE 3

Compute for Cpk measure of process capability

for this machine and interpret your results.

What value would you have obtained with Cp

measure?

USL = 100 LSL = 60

Process σ = 4 process mean = 80

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SAMPLE 4 Food served at a restaurant should be between 38°C and 49°C when it is delivered to the customer. The process used to keep the food at the correct temperature has a process standard deviation of 2°C and the mean value for these temperature is 40. What is the process capability of the process and the capability process index?

Given,

USL (Upper Specification Limit) =49°C

LSL (Lower Specification Limit) =38°C

Standard Deviation =2°C Mean = 40

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statistical quality control telephone

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